I try to log in with ssh passwordless into a server that I've set up. sshd
is running there. I can log in with password.
When I set up the crypto (ssh-keygen -t ecdsa) and install the pub key at
the host, I can't login anymore, not by defaulting to password either.

How did you configure ssh? Did you run ssh-user-config?
Does it work with the default rsa key?

Note that I do have success with different Windows XP and Vista
machines. It's only this one Windows 7 case that's misbehaving. And the
problem is independent whether I log in from another cygwin or a linux
machine. It's always that one server under W7.

I'd very much like to inspect logfiles on the server's side. Despite
that I've arranged for sshd-stdout.log sshd-stderr.log files in /var/log
(they are created), and despite that I have installed and activated the
syslog-ng package the logins, whether failing or succeeding (keys
disabled and password dialog), leave no messages in any of the logfiles.
( cygrunsrv --install sshd --path /usr/sbin/sshd -1 /var/log/sshd-1.log
-2 /var/log/sshd-2.log --args -e; cygrunsrv -S sshd )
I've also looked into the W7 event logs, in vain.

Other clues for narrowing in: I connect within the same IP-domain,
machines sit side-by-side, I give the full address
HGS@W7machine.oso.chalmers.se
and HGS is the user exactly as specified in /etc/passwd
Can the firewall prevent receiving the send_publickey_test ? Isn't
send_publickey_test going thru port 22 too ?
ssh -i .ssh/id_ecdsa.pub HGS@W7machine.oso.chalmers.se
Any option I miss in the ssh command line?